About static output controller existing

The problem of the existence of static output controllers for linear time-invariant continuous-time controlled and observed plants in the general case is considered and its solution is given. The obvious advantage of static output feedback control over state control is that it does not require measuring all state variables to implement it. An equally significant advantage over the control in the form of a linear dynamic controller is that the dimensions of the closed-loop and initial plants are equal. It is shown that by reducing the input and output matrices to a block-homogeneous form, the initial bilinear inequality with respect to the matrix of the Lyapunov quadratic function and the controller matrix can be represented as a single block symmetric matrix. Thanks to this, it is possible to formulate the necessary and sufficient conditions for the existence of static output feedback. In accordance with the presented theorem, it is concluded that if the matrix of an object can be divided into blocks in such a way that at least one of its diagonal blocks was Hurwitz, then a static regulator exists. If this condition is met, then for the existence of a static output controller, it is necessary and sufficient that there is a state controller for an unstable diagonal block of the object matrix. The obtained results allow us to formulate criteria by which conclusions can be drawn regarding the possibility of static stabilization of a given linear object. Examples are presented, which demonstrate the application of the results obtained.